Polyakov loop at next-to-next-to-leading order

We calculate the next-to-next-to-leading correction to the expectation value of the Polyakov loop or equivalently to the free energy of a static charge. This correction is of order $g^5$. We show that up to this order the free energy of the static charge is proportional to the quadratic Casimir operator of the corresponding representation. We also compare our perturbative result with the most recent lattice results in SU(3) gauge theory.

Figure 1

All Feynman diagrams contributing to ${\mathrm{\Pi }}_{m_D}^{(2)}(0,k\sim m_D)$. Dashed lines represent temporal gluons, curly lines spatial gluons, and dotted lines with arrows are ghost propagators. The diagrams are labeled $L_1,\dots ,L_{12}$ from top-left to bottom-right. $L_7,\dots ,L_{12}$ are self-energy insertions into one-loop diagrams, while $L_1,\dots ,L_6$ are new two-loop configurations.

Figure 2

All $\mathcal{O}(g^5)$ diagrams that can give a contribution from the scale $m_M$. The bubble stands for the resummed propagator.

Figure 3

Connected three-gluon diagrams.

Figure 4

Diagrams at $\mathcal{O}(g^8)$ with a color coefficient $C_R^{(4)}$.

Figure 5

The free energy of a static quark $F_Q$ for the SU(3) gauge theory in weak coupling expansion at LO, NLO and NNLO. The bands are obtained by varying the renormalization scale $\mu$ between $\pi T$ and $6\pi T$. Also shown are the lattice data for $F_Q$ obtained on lattices with temporal extent $N_{\tau }=4$ and 8 [20].

Figure 6

The entropy of a static quark $S_Q$ for the SU(3) gauge theory in weak coupling expansion at LO, NLO and NNLO. The bands are obtained by varying the renormalization scale $\mu$ between $\pi T$ and $6\pi T$. Also shown are the lattice results for $S_Q$; cf. the description in the text.

Figure 7

EQCD type and mixed contributions to $F_Q$ at $\mathcal{O}(g^5)$ (upper panel) and $\mathcal{O}(g^6)$ (lower panel). The bands correspond to the variation of the renormalization scale from $\pi T$ to $6\pi T$. The thick black line corresponds to the higher order EQCD type contributions from lattice EQCD estimated in Ref. [50] for the renormalization scale $\mu =4\pi T$; cf. the description in the text.

Figure 8

All free propagators and interaction vertices in phase-space Coulomb gauge. Whenever there is an arrow specifying the direction of a momentum over a mixed propagator, opposite momenta will give the negative propagator.

Figure 9

All diagrams relevant for the cancellation of the nonzero modes in the one-loop spatial gluon self-energy in Coulomb gauge.

Figure 10

All two-loop diagram topologies in phase-space Coulomb gauge. The double-line propagators can represent either a temporal, an electric, or a mixed propagator. Also the diagram with two one-loop bubbles is displayed in the bottom-right corner. We will label the diagrams ${\tilde{L}}_1,\dots ,{\tilde{L}}_7$ from top left to bottom right.

Figure 11

Explicit expression for a double line propagator with two vertices in terms of temporal, electric, and mixed propagators.

Figure 12

Additional diagrams carrying the scale $m_M$ at $\mathcal{O}(g^6)$.

Figure 13

All unconnected three-gluon diagrams. The corresponding color coefficients are labeled $C_{ij}$, where $i$ denotes the row and $j$ the column in which the diagram is listed.